Suppose a matter in a generic curved spacetime is given by a density matrix [1]}, [2]} \(\hat{\rho }\)  which could be either a pure or a mixed state. Then we obtain the following relation
\(\begin{split}\langle :\hat{T}_{\mu \nu }:n^{\mu }n^{\nu }\rangle & =\text{Tr}[\hat{\rho }:\hat{T}_{\mu \nu }(x):n^{\mu }n^{\nu }]=\sum _{j}\mathinner {\langle {\mathcal {O}^{(j)}}|}\hat{\rho }:\hat{T}_{\mu \nu }(x):n^{\mu }n^{\nu }\mathinner {|{\mathcal {O}^{(j)}}\rangle }\\& =\sum _{i,j}\mathinner {\langle {\mathcal {O}^{(j)}}|}\hat{\rho }\mathinner {|{\mathcal {O}^{(i)}}\rangle }\mathinner {\langle {\mathcal {O}^{(i)}}|}:\hat{T}_{\mu \nu }(x):n^{\mu }n^{\nu }\mathinner {|{\mathcal {O}^{(j)}}\rangle }\\& =\sum _{i,j}\Big [\mathinner {\langle {\mathcal {O}^{(j)}}|}\hat{\rho }\mathinner {|{\mathcal {O}^{(i)}}\rangle }\mathcal {N}(F_{ij},F_{ij})e^{-\frac{1}{2}||\mathcal {O}^{(i)}-\mathcal {O}^{(j)}||^{2}}e^{i\Delta (\mathcal {O}^{(i)},\mathcal {O}^{(j)})}\Big ],\end{split}\) 
